For any sets A and B, give an example of a one-to-one correspondence A times B r

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One such correspondence would be, for any element (a,b) in AxB (where a is in A, and b is in B): f: (a,b) ---> (b,a) First, we check that this is indeed from AxB to BxA. Since b is in B and a is in A, it indeed is from AxB to BxA. Second, is this one-to-one? We need to check that two points in AxB correspond to the same point in BxA ONLY when they two points are in fact equal. Suppose f[(a,b)] = f[(a',b')]. Then (b,a) = (b',a'), and so b=b' and a=a'. However, if b=b' and a=a', then the two points we started with: (a,b) and (a',b') were the same point. Therefore any two points in AxB that correspond to the SAME point in BxA must indeed be the same points (in AxB). Finally, is this onto/surjective? We need to check that any element in BxA is the image under f of SOME point in AxB. Let (b,a) be a generic element of BxA, with b in B and a in A. But (b,a) = f[(a,b)], and so each element of BxA indeed corresponds with an element of AxB. Alternatively: You could show that f is a one-to-one correspondence by showing that g: BxA ---> AxB via g[(b,a)] = (a,b) is a (two-sided) inverse. However, the above argument is the more straightforward, I believe. Hope this helps.

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